Approximate analytical solutions of systems of PDEs by homotopy analysis method

In this paper, the homotopy analysis method (HAM) is applied to obtain series solutions to linear and nonlinear systems of first- and second-order partial differential equations (PDEs). The HAM solutions contain an auxiliary parameter which provides a convenient way of controlling the convergence region of series solutions. It is shown in particular that the solutions obtained by the variational iteration method (VIM) are only special cases of the HAM solutions.

[1]  I. Pop,et al.  Explicit analytic solution for similarity boundary layer equations , 2004 .

[2]  S. Liao.  An approximate solution technique not depending on small parameters: A special example , 1995 .

[3]  A. Bataineh,et al.  On a new reliable modification of homotopy analysis method , 2009 .

[4]  Shijun Liao,et al.  On the homotopy analysis method for nonlinear problems , 2004, Appl. Math. Comput..

[5]  Mohd. Salmi Md. Noorani,et al.  The homotopy analysis method for Cauchy reaction–diffusion problems , 2008 .

[6]  Mohd. Salmi Md. Noorani,et al.  Series solution of the multispecies Lotka-Volterra equations by means of the homotopy analysis method , 2008 .

[7]  S. Liao.  AN EXPLICIT TOTALLY ANALYTIC APPROXIMATION OF BLASIUS VISCOUS FLOW PROBLEMS , 1999 .

[8]  Abdul-Majid Wazwaz,et al.  The variational iteration method for solving linear and nonlinear systems of PDEs , 2007, Comput. Math. Appl..

[9]  Shijun Liao,et al.  Comparison between the homotopy analysis method and homotopy perturbation method , 2005, Appl. Math. Comput..

[10]  Mohd. Salmi Md. Noorani,et al.  Solutions of time-dependent Emden–Fowler type equations by homotopy analysis method , 2007 .

[11]  Allen T. Chwang,et al.  Series solutions for a nonlinear model of combined convective and radiative cooling of a spherical body , 2006 .

[12]  A. Bataineh,et al.  Modified homotopy analysis method for solving systems of second-order BVPs , 2009 .

[13]  S. Abbasbandy,et al.  Homotopy analysis method for quadratic Riccati differential equation , 2008 .

[14]  Shijun Liao,et al.  AN APPROXIMATE SOLUTION TECHNIQUE WHICH DOES NOT DEPEND UPON SMALL PARAMETERS (PART 2): AN APPLICATION IN FLUID MECHANICS , 1997 .

[15]  Santanu Saha Ray,et al.  A numerical solution of the coupled sine-Gordon equation using the modified decomposition method , 2006, Appl. Math. Comput..

[16]  Ji-Huan He,et al.  Homotopy perturbation method: a new nonlinear analytical technique , 2003, Appl. Math. Comput..

[17]  Ebrahimpour,et al.  Approximate explicit solutions of nonlinear BBMB equations by homotopy analysis method and comparison with the exact solution , 2007 .

[18]  S. Abbasbandy.  Approximate solution for the nonlinear model of diffusion and reaction in porous catalysts by means of the homotopy analysis method , 2008 .

[19]  S. Liao,et al.  Beyond Perturbation: Introduction to the Homotopy Analysis Method , 2003 .

[20]  S. Liao.  A kind of approximate solution technique which does not depend upon small parameters — II. An application in fluid mechanics , 1997 .

[21]  S. Liao.  An explicit, totally analytic approximate solution for Blasius’ viscous flow problems , 1999 .

[22]  S. Abbasbandy.  THE APPLICATION OF HOMOTOPY ANALYSIS METHOD TO NONLINEAR EQUATIONS ARISING IN HEAT TRANSFER , 2006 .

[23]  S. Abbasbandy.  The application of homotopy analysis method to solve a generalized Hirota–Satsuma coupled KdV equation , 2007 .

[24]  T. Hayat,et al.  Homotopy analysis of MHD flows of an Oldroyd 8-constant fluid , 2004 .

[25]  S. Liao.  A new branch of solutions of boundary-layer flows over an impermeable stretched plate , 2005 .

[26]  T. Hayat,et al.  Homotopy Solutions for a Generalized Second-Grade Fluid Past a Porous Plate , 2005 .

[27]  Tasawar Hayat,et al.  Exact flow of a third grade fluid past a porous plate using homotopy analysis method , 2003 .

[28]  Abdul-Majid Wazwaz,et al.  Partial differential equations : methods and applications , 2002 .

[29]  Shijun Liao,et al.  AN APPROXIMATE SOLUTION TECHNIQUE WHICH DOES NOT DEPEND UPON SMALL PARAMETERS: A SPECIAL EXAMPLE , 1995 .

[30]  Mohd Salmi Md Noorani,et al.  Numerical simulations of systems of PDEs by variational iteration method , 2008 .

[31]  A. Bataineh,et al.  Solving systems of ODEs by homotopy analysis method , 2008 .